Page 177 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 159

Cov { F(k),F(t)) = E { F(k)F* (9) E ( F(k) )E ( F*(&)
)
-

=O for k+L. (4.1 18)

That is, F(k) and F(L) are uncorrelated. It means that the covariance matrices
of X for all classes are simultaneously diagonalized by the Fourier transform, if
the random processes are stationary.

The quadratic classifier in the Fourier domain: Thus, if the F(k)'s are
normally distributed, we can design a quadratic classifier in the Fourier domain
as

(4.1 19)
+ I),, 9

where

(4.120)

(4.121)

Note in (4.119) that, since the covariance matrices of the F(j)'s for both w1
and w2 are diagonal, all cross terms between F(j) and F(k) disappear.
A modification of the quadratic classifier can be made by treating (4.119)
as a linear classifier and finding the optimum vI,, v2,, and vo instead of using
(4.120) and (4.121). In this approach, we need to optimize only 2n+l parame-
ters, v,, (i = 1,2; j = 0, . . . ,n-l) and vo.